Principles of electromagnetics. 1, Understanding vectors & electrostatic fields

108 Pages·2015·14.27 MB·English

by  Adams

Principles of
Electromagnetics
1—Understanding
Vectors &
Electrostatic
Fields
Arlon T. Adams
Jay K. Lee
9781606507155_Cover.indd 1 19/12/14 3:45 PM
Principles of
Electromagnetics 1—
Understanding Vectors &
Electrostatic Fields
Arlon T. Adams
Jay K. Lee
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Electromagnetics 1—Understanding Vectors & Electrostatic Fields
Copyright © Cognella Academic Publishing 2015
www.cognella.com
All rights reserved. No part of this publication may be reproduced,
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except for brief quotations, not to exceed 400 words, without the prior
permission of the publisher.
ISBN-13: 978-1-60650-715-5 (e-book)
www.momentumpress.net
Trademark Notice: Product or corporate names may be trademarks
or registered trademarks, and are used only for identification and
explanation without intent to infringe.
A publication in the Momentum Press Electrical Power collection
Cover and interior design by S4Carlisle Publishing Services Private Ltd.,
Chennai, India
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Brief Contents
Preface .................................................................................................vii
Chapter 1 Introduction to Vectors .....................................................1
Chapter 2 Introduction to Electrostatic Fields and
Electromagnetic Potentials ..............................................47
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Contents
Chapter 1 Introduction to Vectors .....................................................1
1.1 Introduction ............................................................1
1.1.1 Josiah Willard Gibbs (1839-1903) and the
Development of Vector Analysis ........1
1.2 VECTOR ALGEBRA ..............................................4
1.2.1 Basic Operations of Vector Algebra ...................4
1.2.2 Vector Algebra in Rectangular Coordinates .......7
1.2.3 Triple Products .................................................9
1.3 COORDINATE SYSTEMS ..................................12
1.3.1 Coordinate System Geometry .........................12
1.3.2 Differential Elements of Length, Surface
and Volume .......................14
1.3.3 Coordinate Transformations ...........................17
1.3.4 Integrals of Vector Functions ..........................21
1.4 VECTOR CALCULUS .........................................26
1.4.1 Definitions .....................................................26
1.4.2 Gradient .........................................................27
1.4.3 Divergence .....................................................31
1.4.5 The Divergence Theorem and Stokes’
Theorem – Solenoidal and Conservative
Fields .................................34
1.4.6 Vector Identities .............................................41
1.4.7 Higher Order Functions of Vector Calculus ....43
1.5 HELMHOLTZ’S THEOREM ..............................44
Chapter 2 Introduction to Electrostatic Fields and
Electromagnetic Potentials ..............................................47
2.1 Introduction ..........................................................47
2.2 Electric Charge .......................................................48
2.3 The Electric Field in Free Space ..............................52
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2.4 Charles Augustin Coulomb (1736-1806) and
the Discovery of Coulomb’s Law ............................54
2.5 Gauss’ Law .............................................................60
2.6. The Electric Fields Of Arbitrary C
harge Distributions ...............................................67
2.7. The Scalar Electric Potential V ..............................75
2.8. Potential of an Arbitrary Charge Distribution .......77
2.9. CONDUCTORS .................................................83
2.10. The Electric Dipole ...............................................89
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List of Figures
Figure 1-1. The vector A as a directed line segment. ............................4
Figure 1-2. Addition of vectors. ...........................................................5
Figure 1-4. The cross product. .............................................................6
Figure 1-3. The dot product. ...............................................................6
Figure 1-5. Representation of a vector in rectangular coordinate
system. ..............................................................................7
Figure 1-6. The Triple Product A · (B × C). .......................................10
Figure 1-7. The three basic coordinate systems. .................................12
Figure 1-8. Orthogonal surfaces and unit vectors. .............................14
Figure 1-9. Basic elements of differential length in cylindrical
and spherical coordinates. ...............................................15
Figure 1-10. Basic surface elements. ....................................................16
Figure 1-11. The transformation between rectangular and
cylindrical coordinates. ....................................................17
Figure 1-12. The transformation between cylindrical and spherical
coordinates. .....................................................................18
Figure 1-13. Line integrals. .................................................................22
Figure 1-14. Independence of path. .....................................................24
Figure 1-15. Surface integrals. .............................................................25
Figure 1-16. Temperature (T) and temperature gradient (DT) in a
room. ..............................................................................28
Figure 1-17. The definition of divergence and curl. .............................31
Figure 1-18. A vector field that has divergence and curl. ......................32
Figure 1-19. A hemispherical volume. .................................................35
Figure 1-20. A semi-circular contour. ..................................................37
Figure 1-21. The electric field of a point charge. ..................................38
Figure 1-22. The magnetic field of a current filament. .........................40
Figure 2-1. Electric charge distribution (a) Volume charge
density ρv. (b) Surface charge density ρs. (c) Line
charge density ρℓ. (d) A point charge q. ..........................50
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Figure 2-2. Coulomb’s law. ................................................................59
Figure 2-3. Spherical charge distributions. .........................................61
Figure 2-4. Cylindrical and planar charge distributions. ....................64
Figure 2-5. A uniform line charge density ρℓo of finite length. ..........67
Figure 2-6. An arbitrary volume charge distribution ρv(x′, y′, z′)
in the basic source point-field point representation..........68
Figure 2-7. A uniformly charged disk. ...............................................70
Figure 2-8. A uniform line charge density of finite length..................73
Figure 2-9. Parallel plates with applied voltage. .................................76
Figure 2-10. Electric potential of a point charge q (a) at the origin,
and (b) at the source point (x′, y′, z′). .............................76
Figure 2-11(a). A symmetric surface charge distribution for a
spherical conductor. ........................................................83
Figure 2-11(b). A surface charge distribution for an arbitrarily-
shaped conductor. ...........................................................84
Figure 2-11(c). A surface charge for a conductor in the presence of
an applied field. ...............................................................84
Figure 2-12(a). A charged conductor. ..................................................85
Figure 2-12(b). An uncharged conductor with charge source nearby. ...85
Figure 2-13(a). A charged hollow conductor. .......................................86
Figure 2-13(b). A charged hollow conductor with source inside. .........87
Figure 2-14. An air-conductor interface. .............................................87
Figure 2-15. A point charge within a conducting shell. .......................88
Figure 2-16. A dipole. .........................................................................90
Figure 2-17 (a). A quadrupole. ............................................................92
Figure 2-17(b). A linear quadrupole. ...................................................92
Figure 2-18. An octopole. ...................................................................92
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Preface
Electromagnetics is not an easy subject for students. The subject presents
a number of challenges, such as: new math, new physics, new geometry,
new insights and difficult problems. As a result, every aspect needs to be
presented to students carefully, with thorough mathematics and strong
physical insights and even alternative ways of viewing and formulating
the subject. The theoretician James Clerk Maxwell and the experimental-
ist Michael Faraday, both shown on the cover, had high respect for physi-
cal insights.
This book is written primarily as a text for an undergraduate course
in electromagnetics, taken by junior and senior engineering and phys-
ics students. The book can also serve as a text for beginning graduate
courses by including advanced subjects and problems. The book has been
thoroughly class-tested for many years for a two-semester Electromagnet-
ics course at Syracuse University for electrical engineering and physics
students. It could also be used for a one-semester course, covering up
through Chapter 8 and perhaps skipping Chapter 4 and some other parts.
For a one-semester course with more emphasis on waves, the instructor
could briefly cover basic materials from statics (mainly Chapters 2 and 6)
and then cover Chapters 8 through 12.
The authors have attempted to explain the difficult concepts of elec-
tromagnetic theory in a way that students can readily understand and
follow, without omitting the important details critical to a solid under-
standing of a subject. We have included a large number of examples, sum-
mary tables, alternative formulations, whenever possible, and homework
problems. The examples explain the basic approach, leading the students
step by step, slowly at first, to the conclusion. Then special cases and
limiting cases are examined to draw out analogies, physical insights and
their interpretation. Finally, a very extensive set of problems enables the
instructor to teach the course for several years without repeating problem
assignments. Answers to selected problems at the end allow students to
check if their answers are correct.
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